Número de lançamentos na mesa de dados

Introdução

Uma pergunta que me fazem com frequência é: "Qual a probabilidade de um jogador sobreviver a x rodadas no craps?" A tabela a seguir responde a essa pergunta para até 50 rodadas. A primeira coluna indica o número da rodada. A segunda coluna indica a probabilidade de sair um sete (ou sete-out) exatamente nessa rodada. A terceira coluna indica a probabilidade de sobreviver APÓS essa rodada.

Tabela de Sobrevivência do Craps para 1 a 50 Lançamentos

Rolar Número	Probabilidade Sete-Out	Probabilidade Sobrevivência
1	0,00000000	1.00000000
2	0,11111111	0,88888889
3	0,11676955	0,77211934
4	0,10476680	0,66735254
5	0,09122363	0,57612891
6	0,07891804	0,49721087
7	0,06816676	0,42904411
8	0,05885276	0,37019135
9	0,05080065	0,31939070
10	0,04384414	0,27554656
11	0,03783614	0,23771043
12	0,03264850	0,20506193
13	0,02817002	0,17689190
14	0,02430433	0,15258757
15	0,02096801	0,13161956
16	0,01808886	0,11353070
17	0,01560445	0,09792625
18	0,01346084	0,08446541
19	0,01161138	0,07285403
20	0,01001580	0,06283823
21	0,00863931	0,05419892
22	0,00745187	0,04674705
23	0,00642755	0,04031950
24	0,00554396	0,03477554
25	0,00478180	0,02999374
26	0,00412437	0,02586937
27	0,00355731	0,02231206
28	0,00306819	0,01924387
29	0,00264632	0,01659755
30	0,00228244	0,01431511
31	0,00196858	0,01234653
32	0,00169788	0,01064864
33	0,00146440	0,00918424
34	0,00126303	0,00792121
35	0,00108934	0,00683187
36	0,00093954	0,00589234
37	0,00081033	0,00508201
38	0,00069890	0,00438311
39	0,00060278	0,00378033
40	0,00051989	0,00326044
41	0,00044839	0,00281205
42	0,00038673	0,00242532
43	0,00033354	0,00209178
44	0,00028767	0,00180411
45	0,00024811	0,00155600
46	0,00021399	0,00134201
47	0,00018456	0,00115745
48	0,00015918	0,00099827
49	0,00013729	0,00086098
50	0,00011841	0,00074257

A próxima tabela mostra as mesmas informações, mas para até 200 lançamentos. As probabilidades ficam muito pequenas, então esta tabela está em notação científica.

Tabela de Sobrevivência do Craps para 1 a 200 Lançamentos

Rolar Número	Probabilidade Sete-Out	Probabilidade Sobrevivência
1	0,000000E+00	1.000000E+00
2	1.111111E-01	8.888889E-01
3	1.167695E-01	7.721193E-01
4	1.047668E-01	6.673525E-01
5	9.122363E-02	5.761289E-01
6	7.891804E-02	4.972109E-01
7	6.816676E-02	4.290441E-01
8	5.885276E-02	3.701913E-01
9	5.080065E-02	3.193907E-01
10	4.384414E-02	2.755466E-01
11	3,783614E-02	2.377104E-01
12	3.264850E-02	2.050619E-01
13	2.817002E-02	1,768919E-01
14	2.430433E-02	1.525876E-01
15	2.096801E-02	1.316196E-01
16	1.808886E-02	1.135307E-01
17	1.560445E-02	9,792625E-02
18	1.346084E-02	8.446541E-02
19	1.161138E-02	7.285403E-02
20	1.001580E-02	6.283823E-02
21	8.639309E-03	5.419892E-02
22	7.451869E-03	4.674705E-02
23	6.427548E-03	4.031950E-02
24	5.543963E-03	3.477554E-02
25	4.781795E-03	2.999374E-02
26	4.124373E-03	2.586937E-02
27	3.557310E-03	2.231206E-02
28	3.068195E-03	1.924387E-02
29	2,646317E-03	1,659755E-02
30	2.282437E-03	1.431511E-02
31	1,968585E-03	1.234653E-02
32	1,697884E-03	1.064864E-02
33	1.464404E-03	9.184241E-03
34	1.263027E-03	7.921214E-03
35	1.089340E-03	6.831874E-03
36	9.395362E-04	5.892338E-03
37	8.103321E-04	5.082006E-03
38	6.988952E-04	4.383111E-03
39	6.027824E-04	3.780328E-03
40	5.198867E-04	3.260442E-03
41	4.483907E-04	2.812051E-03
42	3,867267E-04	2.425324E-03
43	3.335427E-04	2.091782E-03
44	2,876726E-04	1.804109E-03
45	2.481107E-04	1.555998E-03
46	2.139894E-04	1.342009E-03
47	1,845605E-04	1.157448E-03
48	1.591789E-04	9.982695E-04
49	1,372878E-04	8.609818E-04
50	1.184072E-04	7.425745E-04
51	1.021232E-04	6.404513E-04
52	8.807867E-05	5.523726E-04
53	7.596559E-05	4.764071E-04
54	6.551837E-05	4.108887E-04
55	5.650790E-05	3.543808E-04
56	4,873660E-05	3.056442E-04
57	4.203405E-05	2,636101E-04
58	3,625328E-05	2.273569E-04
59	3.126751E-05	1.960893E-04
60	2,696741E-05	1,691219E-04
61	2.325869E-05	1.458632E-04
62	2.006001E-05	1.258032E-04
63	1.730124E-05	1.085020E-04
64	1,492187E-05	9.358012E-05
65	1,286972E-05	8.071040E-05
66	1.109980E-05	6.961061E-05
67	9.573283E-06	6.003732E-05
68	8.256706E-06	5.178062E-05
69	7.121193E-06	4.465942E-05
70	6.141842E-06	3,851758E-05
71	5.297178E-06	3.322040E-05
72	4.568677E-06	2,865173E-05
73	3.940364E-06	2.471136E-05
74	3.398461E-06	2.131290E-05
75	2.931083E-06	1,838182E-05
76	2.527982E-06	1,585384E-05
77	2.180319E-06	1,367352E-05
78	1,880468E-06	1.179305E-05
79	1,621854E-06	1.017120E-05
80	1.398806E-06	8.772390E-06
81	1.206434E-06	7.565956E-06
82	1.040518E-06	6.525439E-06
83	8.974191E-07	5.628020E-06
84	7.740004E-07	4.854019E-06
85	6.675550E-07	4.186464E-06
86	5.757487E-07	3.610715E-06
87	4,965681E-07	3.114147E-06
88	4.282770E-07	2.685870E-06
89	3,693777E-07	2.316493E-06
90	3.185785E-07	1.997914E-06
91	2,747656E-07	1,723148E-06
92	2,369781E-07	1.486170E-06
93	2.043874E-07	1,281783E-06
94	1,762788E-07	1.105504E-06
95	1.520358E-07	9.534683E-07
96	1.311269E-07	8.223414E-07
97	1.130935E-07	7.092478E-07
98	9.754019E-08	6.117076E-07
99	8.412586E-08	5.275818E-07
100	7.255634E-08	4.550254E-07
101	6.257794E-08	3,924475E-07
102	5.397183E-08	3,384757E-07
103	4.654929E-08	2.919264E-07
104	4.014754E-08	2.517788E-07
105	3.462620E-08	2.171526E-07
106	2.986419E-08	1.872885E-07
107	2.575708E-08	1.615314E-07
108	2.221480E-08	1,393166E-07
109	1.915969E-08	1.201569E-07
110	1,652473E-08	1.036322E-07
111	1.425214E-08	8.938002E-08
112	1.229210E-08	7.708792E-08
113	1.060161E-08	6.648631E-08
114	9.143612E-09	5.734269E-08
115	7,886126E-09	4,945657E-08
116	6.801576E-09	4.265499E-08
117	5,866181E-09	3,678881E-08
118	5.059427E-09	3.172938E-08
119	4.363623E-09	2,736576E-08
120	3,763510E-09	2.360225E-08
121	3.245929E-09	2.035632E-08
122	2,799529E-09	1,755679E-08
123	2.414520E-09	1.514227E-08
124	2.082460E-09	1.305981E-08
125	1,796067E-09	1,126375E-08
126	1.549061E-09	9.714685E-09
127	1.336024E-09	8.378661E-09
128	1.152286E-09	7.226375E-09
129	9,938163E-10	6.232559E-09
130	8,571405E-10	5.375419E-09
131	7,392611E-10	4.636157E-09
132	6,375933E-10	3.998564E-09
133	5,499075E-10	3,448657E-09
134	4,742808E-10	2,974376E-09
135	4,090547E-10	2,565321E-09
136	3,527990E-10	2.212522E-09
137	3,042799E-10	1.908242E-09
138	2,624334E-10	1,645809E-09
139	2,263419E-10	1.419467E-09
140	1,952140E-10	1,224253E-09
141	1,683669E-10	1.055886E-09
142	1,452120E-10	9.106740E-10
143	1,252416E-10	7,854324E-10
144	1,080176E-10	6,774148E-10
145	9,316232E-11	5,842525E-10
146	8.035006E-11	5,039024E-10
147	6,929981E-11	4,346026E-10
148	5,976927E-11	3,748334E-10
149	5,154943E-11	3,232839E-10
150	4,446003E-11	2,788239E-10
151	3,834561E-11	2,404783E-10
152	3.307208E-11	2,074062E-10
153	2,852380E-11	1,788824E-10
154	2,460103E-11	1,542814E-10
155	2,121774E-11	1,330637E-10
156	1,829975E-11	1,147639E-10
157	1,578305E-11	9,898086E-11
158	1,361247E-11	8,536839E-11
159	1,174039E-11	7,362800E-11
160	1,012578E-11	6.350222E-11
161	8,733222E-12	5,476899E-11
162	7,532174E-12	4,723682E-11
163	6,496303E-12	4,074052E-11
164	5.602891E-12	3,513763E-11
165	4,832346E-12	3,030528E-11
166	4,167772E-12	2,613751E-11
167	3,594594E-12	2,254292E-11
168	3,100243E-12	1,944267E-11
169	2,673878E-12	1,676880E-11
170	2,306150E-12	1,446265E-11
171	1,988993E-12	1,247365E-11
172	1,715455E-12	1,075820E-11
173	1,479535E-12	9,278663E-12
174	1,276060E-12	8.002603E-12
175	1,100568E-12	6,902035E-12
176	9.492110E-13	5,952824E-12
177	8,186696E-13	5,134155E-12
178	7,060810E-13	4,428074E-12
179	6,089764E-13	3,819097E-12
180	5,252261E-13	3,293871E-12
181	4,529938E-13	2,840877E-12
182	3,906952E-13	2,450182E-12
183	3,369644E-13	2.113218E-12
184	2,906229E-13	1,822595E-12
185	2,506546E-13	1,571940E-12
186	2,161831E-13	1,355757E-12
187	1,864522E-13	1,169305E-12
188	1,608101E-13	1,008495E-12
189	1,386945E-13	8,698004E-13
190	1,196204E-13	7.501800E-13
191	1,031694E-13	6.470106E-13
192	8.898094E-14	5,580296E-13
193	7,674372E-14	4,812859E-13
194	6,618945E-14	4,150965E-13
195	5,708666E-14	3,580098E-13
196	4,923575E-14	3,087741E-13
197	4,246454E-14	2,663095E-13
198	3,662455E-14	2,296850E-13
199	3,158771E-14	1,980973E-13
200	2,724357E-14	1,708537E-13

O número médio de lançamentos por atirador é 8,525510.

Para saber como resolvi este problema, consulte meu site MathProblems.info , problema 204.

Escrito por: Michael Shackleford